Optimal. Leaf size=99 \[ -2 d p x-\frac {1}{2} e p x^2+\frac {2 \sqrt {a} d p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {\left (b d^2-a e^2\right ) p \log \left (a+b x^2\right )}{2 b e}+\frac {(d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 e} \]
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Rubi [A]
time = 0.05, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2513, 815, 649,
211, 266} \begin {gather*} \frac {2 \sqrt {a} d p \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {(d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 e}-\frac {p \left (b d^2-a e^2\right ) \log \left (a+b x^2\right )}{2 b e}-2 d p x-\frac {1}{2} e p x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 266
Rule 649
Rule 815
Rule 2513
Rubi steps
\begin {align*} \int (d+e x) \log \left (c \left (a+b x^2\right )^p\right ) \, dx &=\frac {(d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 e}-\frac {(b p) \int \frac {x (d+e x)^2}{a+b x^2} \, dx}{e}\\ &=\frac {(d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 e}-\frac {(b p) \int \left (\frac {2 d e}{b}+\frac {e^2 x}{b}-\frac {2 a d e-\left (b d^2-a e^2\right ) x}{b \left (a+b x^2\right )}\right ) \, dx}{e}\\ &=-2 d p x-\frac {1}{2} e p x^2+\frac {(d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 e}+\frac {p \int \frac {2 a d e-\left (b d^2-a e^2\right ) x}{a+b x^2} \, dx}{e}\\ &=-2 d p x-\frac {1}{2} e p x^2+\frac {(d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 e}+(2 a d p) \int \frac {1}{a+b x^2} \, dx+\frac {\left (\left (-b d^2+a e^2\right ) p\right ) \int \frac {x}{a+b x^2} \, dx}{e}\\ &=-2 d p x-\frac {1}{2} e p x^2+\frac {2 \sqrt {a} d p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {\left (b d^2-a e^2\right ) p \log \left (a+b x^2\right )}{2 b e}+\frac {(d+e x)^2 \log \left (c \left (a+b x^2\right )^p\right )}{2 e}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 83, normalized size = 0.84 \begin {gather*} -2 d p x+\frac {2 \sqrt {a} d p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+d x \log \left (c \left (a+b x^2\right )^p\right )+\frac {1}{2} e \left (-p x^2+\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 93, normalized size = 0.94
method | result | size |
default | \(d \ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) x -2 d p x +\frac {2 d p a \arctan \left (\frac {b x}{\sqrt {b a}}\right )}{\sqrt {b a}}+\frac {e \ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) x^{2}}{2}-\frac {e p \,x^{2}}{2}+\frac {e \ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) a}{2 b}-\frac {a e p}{2 b}\) | \(93\) |
risch | \(\left (\frac {1}{2} e \,x^{2}+d x \right ) \ln \left (\left (b \,x^{2}+a \right )^{p}\right )+\frac {i \pi d \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} x}{2}+\frac {i \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) x^{2} e \pi }{4}-\frac {i \pi e \,x^{2} \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )}{4}+\frac {i \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} x^{2} e \pi }{4}-\frac {i \pi d \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3} x}{2}-\frac {i \pi e \,x^{2} \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3}}{4}-\frac {i \pi d \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \right ) x}{2}+\frac {i \pi d \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right ) x}{2}+\frac {\ln \left (c \right ) e \,x^{2}}{2}-\frac {e p \,x^{2}}{2}-\frac {p \ln \left (\sqrt {-b a}\, x +a \right ) d \sqrt {-b a}}{b}+\frac {p \ln \left (-\sqrt {-b a}\, x +a \right ) d \sqrt {-b a}}{b}+\frac {p \ln \left (\sqrt {-b a}\, x +a \right ) a e}{2 b}+\frac {p \ln \left (-\sqrt {-b a}\, x +a \right ) a e}{2 b}+\ln \left (c \right ) d x -2 d p x\) | \(387\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 83, normalized size = 0.84 \begin {gather*} \frac {1}{2} \, {\left (\frac {4 \, a d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {a e \log \left (b x^{2} + a\right )}{b^{2}} - \frac {x^{2} e + 4 \, d x}{b}\right )} b p + \frac {1}{2} \, {\left (x^{2} e + 2 \, d x\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 206, normalized size = 2.08 \begin {gather*} \left [-\frac {b p x^{2} e - 2 \, b d p \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 4 \, b d p x - {\left (2 \, b d p x + {\left (b p x^{2} + a p\right )} e\right )} \log \left (b x^{2} + a\right ) - {\left (b x^{2} e + 2 \, b d x\right )} \log \left (c\right )}{2 \, b}, -\frac {b p x^{2} e - 4 \, b d p \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 4 \, b d p x - {\left (2 \, b d p x + {\left (b p x^{2} + a p\right )} e\right )} \log \left (b x^{2} + a\right ) - {\left (b x^{2} e + 2 \, b d x\right )} \log \left (c\right )}{2 \, b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 199 vs.
\(2 (92) = 184\).
time = 4.50, size = 199, normalized size = 2.01 \begin {gather*} \begin {cases} \left (d x + \frac {e x^{2}}{2}\right ) \log {\left (0^{p} c \right )} & \text {for}\: a = 0 \wedge b = 0 \\\left (d x + \frac {e x^{2}}{2}\right ) \log {\left (a^{p} c \right )} & \text {for}\: b = 0 \\- 2 d p x + d x \log {\left (c \left (b x^{2}\right )^{p} \right )} - \frac {e p x^{2}}{2} + \frac {e x^{2} \log {\left (c \left (b x^{2}\right )^{p} \right )}}{2} & \text {for}\: a = 0 \\\frac {2 a d p \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{b \sqrt {- \frac {a}{b}}} - \frac {a d \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{b \sqrt {- \frac {a}{b}}} + \frac {a e \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{2 b} - 2 d p x + d x \log {\left (c \left (a + b x^{2}\right )^{p} \right )} - \frac {e p x^{2}}{2} + \frac {e x^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.89, size = 100, normalized size = 1.01 \begin {gather*} \frac {2 \, a d p \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} + \frac {b p x^{2} e \log \left (b x^{2} + a\right ) - b p x^{2} e + 2 \, b d p x \log \left (b x^{2} + a\right ) + b x^{2} e \log \left (c\right ) - 4 \, b d p x + a p e \log \left (b x^{2} + a\right ) + 2 \, b d x \log \left (c\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.13, size = 81, normalized size = 0.82 \begin {gather*} d\,x\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )-\frac {e\,p\,x^2}{2}-2\,d\,p\,x+\frac {e\,x^2\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{2}+\frac {a\,e\,p\,\ln \left (b\,x^2+a\right )}{2\,b}+\frac {2\,\sqrt {a}\,d\,p\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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